Calculate the value of this real+imaginary expression:

[1question]

Homework Statement

Let z = 2 + i and u = -3-3i. Calculate the value of |z/u|.

Homework Equations

None.

The Attempt at a Solution

|z/u|=|(2 + i)/(-3-3i)|
= |(2 + i)/(-3-3i)*(-3+3i/(-3+3i)| --- if it is not clear, I'm multiplying by the conjugate.
= |(-6+6i-3i+3i^2)/(9-9i+9i-9i^2)|
= |(3i^2+3i-6)/(-9i^2+9)|
= |(3i-9)/18|
= |-1/2+i/6|
= 1/2+i/6


Did I take care of the absolute value part correctly? Thanks.

 

[mfb]

The absolute value is a real number, not a complex number. Apart from the last step, it looks fine, but it can be done in an easier way.

 

[1question]

The absolute value is a real number, not a complex number. Apart from the last step, it looks fine, but it can be done in an easier way.

OK, so any hint as to how to convert it to a real number? Also, why does it have to be a real number? What does being in absolute terms have to do with being real/imaginary?

 

[mfb]

OK, so any hint as to how to convert it to a real number?

Check how "absolute value" is defined for complex numbers.

Also, why does it have to be a real number? What does being in absolute terms have to do with being real/imaginary?

Follows from the definition.

 

[micromass]

By the way, it's called a modulus and not an absolute value. Absolute value seems to be reserved for $\mathbb{R}$.

 

[1question]

Ok.

= |-1/2+i/6|
= sqrt((1/2)^2+(1/6)^2)
= srqt(1/4+1/36)
= sqrt(10)/6

Is that it?

 

[SammyS]

Ok.

= |-1/2+i/6|
= sqrt((1/2)^2+(1/6)^2)
= srqt(1/4+1/36)
= sqrt(10)/6

Is that it?

That's correct.

It's also true that |z/u| = |z|/|u| .

$\displaystyle |2+i| = \sqrt{5}$

$\displaystyle |-3-3i| = 3\sqrt{2}$

...

 

[1question]

That's correct.

It's also true that |z/u| = |z|/|u| .

$\displaystyle |2+i| = \sqrt{5}$

$\displaystyle |-3-3i| = 3\sqrt{2}$

...

Thanks for confirming. That IS faster...

 

[Fredrik]

By the way, it's called a modulus and not an absolute value. Absolute value seems to be reserved for $\mathbb{R}$.

I'm pretty sure I've seen it called "absolute value". Let me check... yes, Saff & Snider (Fundamentals of complex analysis for mathematics, science, and engineering) begins the definition by saying "The modulus or absolute value of..." and Anton (Linear Algebra) defines the "modulus" and then immediately says "The modulus of z is also called the absolute value of z". Those are the only books I checked. I also think that "absolute value" is the more common term in physics books.

 

[micromass]

I'm pretty sure I've seen it called "absolute value". Let me check... yes, Saff & Snider (Fundamentals of complex analysis for mathematics, science, and engineering) begins the definition by saying "The modulus or absolute value of..." and Anton (Linear Algebra) defines the "modulus" and then immediately says "The modulus of z is also called the absolute value of z". Those are the only books I checked. I also think that "absolute value" is the more common term in physics books.

I see... I never knew that, I thought modulus was more common. Thanks!